On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems

Multi-frequency, highly-oscillatory Hamiltonian problems derive from the mathematical modelling of many real life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems.Comment: 28 pages, 4 figures (a few typos fixed

High-order energy-conserving Line Integral Methods for charged particle dynamics

In this paper we study arbitrarily high-order energy-conserving methods for simulating the dynamics of a charged particle. They are derived and studied within the framework of Line Integral Methods (LIMs), previously used for defining Hamiltonian Boundary Value Methods (HBVMs), a class of energy-conserving Runge-Kutta methods for Hamiltonian problems. A complete analysis of the new methods is provided, which is confirmed by a few numerical tests.Comment: 24 pages, 3 figures, 2 tables - few typos fixe

Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs

Runge–Kutta–Nyström (RKN) methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered. A general procedure to construct explicit-stage RKN methods with maximal order [fórmula], similar to the developed by the authors (Montijano et al., 2023) for the class of second order IVP under consideration, depending on the nodes [fórmula] is presented. This procedure requires only the solution of successive linear equations in the elementij, [fórmula], of the matrix of coefficients A of the RKN method and avoids the solution of non linear equations. The remarkable fact is that using as free parameters the nodes [fórmula] with a quadrature relation, the [fórmula] elements of matrix can be computed by solving successively linear systems with coefficients depending on the nodes, so that if they are non-singular we get a unique 8-stage method with maximal order [fórmula]. We obtain an optimized six-stage seventh-order RKN method in the sense that the nodes are chosen so that minimize the leading term of the local error. Finally, some numerical experiments are presented to test the behaviour of the optimized RKN method with others with Radau and Lobatto nodes

Algorithm 968: Disode45: A matlab Runge-Kutta solver for piecewise smooth IVPs of Filippov type

In this article, an adaptive Runge-Kutta code, based on the DOPRI5(4) pair for solving initial value problems (IVPs) for differential systems with piecewise smooth solutions (PWS) is presented and the algorithms used in the code are described. The code automatically detects and locates accurately the switching points of the PWS, restarting the integration after each discontinuity. Further, in the case of Filippov systems, algorithms to handle properly sliding mode regimes in an automatic way are included. The code requires the user to provide a description of the IVP and the functions defining the hypersurfaces where the switching points are located, and it returns the discrete approximated solution together with the switching points. Several numerical experiments are presented to illustrate the reliability and efficiency of the code

Space-time spectrally accurate HBVMs for Hamiltonian PDEs

Recently, Hamiltonian Boundary Value Methods (HBVMs), have been used for effectively solving multi-frequency, highly-oscillatory and/or stiffly-oscillatory problems. We here report a few examples showing that, when numerically solving Hamiltonian PDEs, such methods, if coupled with a spectrally accurate space semi-discretization, are able to provide a spectrally accurate solution in time, as well

On the numerical stability of the exponentially fitted methods for first order IVPs

In the numerical solution of Initial Value Problems (IVPs) for differential systems, exponential fitting (EF) techniques are introduced to improve the accuracy behaviour of classical methods when some information on the solutions is known in advance. Typically, these EF methods are evaluated by computing their accuracy for some test problems and it is usual to assume that the stability behaviour is similar to the underlying classical methods. The aim of this paper it to show that for some standard methods the stability behaviour of their exponentially fitted versions may change strongly. Furthermore, this stability depends on the choice of the fitting space, that must be carefully selected in order to assess the quality of the integrators for the IVPs under consideration. In particular, we will show that for the usual fitting space with omega is an element of R the size of the stability domain of the EF method can be much smaller than the one for the original method